1
IIT-JEE 1994
Subjective
+5
-0
Consider the following statements connecting a triangle $$ABC$$

(i) The sides $$a, b, c$$ and area $$\Delta$$ are rational.

(ii) $$a,\tan {B \over 2},\tan {c \over 2}$$ are rational.

(iii) $$a,\sin A,\sin B,\sin C$$ are rational.
Prove that $$\left( i \right) \Rightarrow \left( {ii} \right) \Rightarrow \left( {iii} \right) \Rightarrow \left( i \right)$$

2
IIT-JEE 1994
Subjective
+4
-0
A tower $$AB$$ leans towards west making an angle $$\alpha$$ with the vertical. The angular elevation of $$B$$, the topmost point of the tower is $$\beta$$ as observed from a point $$C$$ due west of $$A$$ at a distance $$d$$ from $$A$$. If the angular elevation of $$B$$ from a point $$D$$ due east of $$C$$ at a distance $$2d$$ from $$C$$ is $$\gamma$$, then prove that $$2$$ tan $$\alpha = - \cot \beta + \cot \gamma$$.
3
IIT-JEE 1993
Subjective
+5
-0
An observer at $$O$$ notices that the angle of elevation of the top of a tower is $${30^ \circ }$$. The line joining $$O$$ to the base of the tower makes an angle of $${\tan ^{ - 1}}\left( {1/\sqrt 2 } \right)$$ with the North and is inclined Eastwards. The observer travels a distance of $$300$$ meters towards the North to a point A and finds the tower to his East. The angle of elevation of the top of the tower at $$A$$ is $$\phi$$, Find $$\phi$$ and the height of the tower.
4
IIT-JEE 1992
Subjective
+4
-0
Three circles touch the one another externally. The tangent at their point of contact meet at a point whose distance from a point of contact is $$4$$. Find the ratio of the product of the radii to the sum of the radii of the circles.
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